Symplectic fixed points, the Calabi invariant and Novikov homology
Symplectic fixed points, the Calabi invariant and Novikov homology
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辛不动点、卡拉比不变量和诺维科夫同调
DOI：
10.1016/0040-9383(94)e0015-c
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
K. Ôno
中科院分区：
文献类型：
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作者：
L. Vân;K. Ôno
流形M上的辛结构w提供了（M, w）的闭1形式与无限小自同构（即满足l . xW= o的向量场X）之间的一一对应关系。如果无限小自同构X对应于一个精确的l~形式，则称为哈密顿向量场。一个在M上的辛形态被称为精确的，如果它是一个时变哈密顿向量场的时变映射。事实上，我们可以找到一个周期哈密顿函数使得rp是哈密顿系统的时间1映射。每个symplectomorphism ep同位素通过symplectomorphisms身份,我们ean分配上同调类卡尔(ep),称为Calabi不变的rp Banyaga [B]表明,rp确切当且仅当卡尔(rp) = o·阿诺德猜想州固定点的数量一个确切symplectomorpbism compaet syrnplectic歧管ean由surn estirnated下面提供的贝蒂的M a11固定点不易变质。阿诺德通过分析rp离恒等式很近的情况得出了这个猜想（参见[AJ]）。如果rp是一个与时无关的哈密顿向量场对应于一个C~-小的莫尔斯函数f的时间1映射，则不动点与f的临界点重合，在这种情况下验证了猜想，这只不过是莫尔斯理论。阿诺德猜想中有许多不完全的结果。Floer取得了很大的进步，他将变分方法（见[CZ]）与Gromov提出的伪<s:1>全纯曲线理论结合起来，证明了单调辛流形的Arnold猜想[Fl]。他为环空间上的泛函作用发展了一个类似莫尔斯理论的理论，并导致了弗洛尔同调的概念。Arnold猜想是由Floer同构于m的普通同构群这一事实推导出来的。最近，Hofer和Salamon [HS]定义了更广泛的一类辛流形（称为弱单调辛流形）的Floer同构群。一个几乎复杂的结构J在M上被w， if校准
A symplectic structure w on a manifold M provides the one-to-one correspondence between closed 1-forms and infinitesimal automorphisms of (M, w), ie vector fields X satisfying L. xW= O. An infinitesimal automorphism X is called a. Hamiltonian vect'or field, if it corresponds to a exact l~ form. A symplectomorphism ep on M is called exact, if it is the time 1 map of a time-depending Hamiltonian vector field. In fact, one can find a. periodic Hamiltonian function such that rp is the time 1 map of the Hamiltonian system. For each symplectomorphism ep isotopic to the identity through symplectomorphisms, oue ean assign a cohomology class Cal (ep), which is called the Calabi invariant of rp Banyaga [B] showed that rp is exact if and only if Cal (rp)= O. The Arnold conjecture states that the number of fixed points of an exact symplectomorpbism on a compaet syrnplectic manifold ean be estirnated below by the surn of the Betti numbers of M provided that a11 the fixed points are non-degenerate. Arnold came to this conjecture by analysing the case that rp is elose to tbe identity (see [AJ). If rp is the time 1 map of a time-independent Hamiltonian vector field corresponding to a Morse function f which is C~-small, the fixed points eoincides with the critical points of fand the conjecture ia verified in this case, which is nothing hut the Morse theory.There are many partial resuIts in the Arnold conjeeture. A great progress was done by Floer, who combined the variational approach (see [CZ]) and theory of pseudüholomorphic curves due to Gromov and proved the Arnold conjeeture for monotone'symplectic manifolds [Fl]. He developed an analogue of the Morse theory for the action funetional on the loop space and led to the notion of Floer homology. The Arnold conjecture is derived from the fact that the Floer homology group is isomorphie to the ordinary homology group of M. Recently, Hofer and Salamon [HS] define the Floer homology group for a wider dass of symplectic manifolds (which are called weakly monotone sympleetie manifolds). An almost complex strueture J on M is calibrated by w, if